Hitchhiking is not recommended. It is not safe.
Hitching should be your LAST option, especially on the remote highways of Patagonia. (Lucky you've got a tent!)
On the other hand, if you get stuck, rural locals are willing to give a lift.
Accommodation in Puerto Natales
Most hikers enjoy a night or two in PN, a small, relaxing tourist town on Last Hope Sound about 110km south of Paine.
Check a guidebook for the latest accommodation options in Puerto Natales. Hostel matrons will meet your bus if you arrange nothing in advance.
Hostels including breakfast start at US$6 / person. (You are unlikely to want to tent there.) Hostels also arrange bus pick-up to Paine for the same cost as you doing it yourself. Very convenient.
Highlights of Puerto Natales? The waterfront is interesting in a bleak Patagonian way.
A number of interesting restaurants cater to Gringos.
PN is certainly adequate for outfitting your hike.
More options for accommodation.
Map of Puerto Natales.
Accommodation in the Park
Aside from hikers accommodation, more expensive hotels are available, offering all manner of luxury services.